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Learning Centre MA 6.21 Measures of Variability (with exercises and answers) Basic Statistics Many courses require that students in undergraduate degrees have a basic understanding of descriptive statistics. Descriptive statistics are statistics that collect, summarize, classify and present data. This handout gives an overview of one type of descriptive statistics, measures of variability. Measures of Variability Measures that allow you to: determine the degree of variation within a population or sample; determine how representative a particular score is of a data set; and determine the scope and validity of any generalizations you wish to make based on your observations in research are measures of variability. The measures of variability discussed in this handout are: Range Variance Standard Deviation Range The range is the difference between the highest and lowest scores in a distribution. It is calculated by subtracting the lowest score from the highest score. EXAMPLE: 14 40 42 47 49 51 71 81 I) Find nouns that are “Specific” (81 – 14) = 67 The range for this distribution would be (81 – 14) = 67. However, as you can see, 14 is an outlier and skews this distribution because if it were not in the distribution, the range would be (81 – 40) = 41. When is the range useful? The range gives you a rough guide to the variability in a data set, as it tells you how a particular score compares to the highest and lowest scores within a data set. The range gives t you only a limited amount of information, as data sets which are skewed towards a low score can have the same range as data sets which are skewed towards a high score, or those which cluster around some central score. H. Williams, 2015 1 Learning Centre MA 6.21 Variability Variance is the degree to which scores vary from their mean. The variance takes into account every score in the data set. The variance is calculated by getting the average of the squared deviations from the mean. To calculate the variance for a set of quiz scores: 1. Find the mean (M). 2. Find the deviation of each raw score from the mean (D). 3. To do this, subtract the mean from each raw score. To check your calculations sum the deviation scores. This sum should be equal to zero. **Note that deviation scores that are below the mean will be negative 4. Square the deviation scores (SS). We do this because by squaring the scores, negative scores are made positive and extreme scores are given relatively more weight. 5. Find the sum of the squared deviation scores. 6. Divide the sum by the number of scores. This yields the average of the squared deviations from the mean, or the variance Range. EXAMPLE: Data Set: 5 6 7 11 12 12 13 14 18 19 21 21 22 34 35 35 46 50 1. 5+6+7+11+12+12+13+14+18+19+21+21+22+24+35+35+35+50 = 360 Mean = 360/N = 360/18 = 20 2. D = N-M 6 – 20 = (-14) 7 – 20 = (-13) 11 – 20 = (-9) 12 – 20 = (-8) 12 – 20 = (-8) 13 – 20 = (-7) 14 – 20 = (-6) 18 – 20 = (-2) 19 – 20 = (-1) 21 – 20 = 1 21 – 20 = 1 22 – 20 = 2 24 – 20 = 4 35 – 20 = 15 35 – 20 = 15 35 – 20 = 15 SS (-15)2 =225 (-14)2 =196 (-13)2 =169 (-9)2 =81 (-8)2 =64 (-8)2 =64 (-7)2 =49 (-6)2 =36 (-2)2 =4 (-1)2 =1 12 =1 12 =1 22 =4 42 =16 152 =225 152 =225 152 =225 50 – 20 = 30 302 =900 Standard 5 –Deviation 20 = (-15) Sum: [(-83) + 83] = 0 2486/18=138.11 Variance = 138.11 H. Williams, 2015 2 Learning Centre MA 6.21 EXERCISE: Using the steps above, find the variance for the following data set. a. 8 11 12 14 17 17 18 19 22 29 35 38 3 Standard Deviation(SD) The standard deviation is the square root of the variance. Unlike the variance, the standard deviation is in the same units as the raw scores themselves, therefore one cannot just use the variance. This is what makes the standard deviation more meaningful. For example, it would make more sense to discuss the variability of a set of IQ scores in IQ points than in squared IQ points because they would not be congruent with the scores meaning. Sum of Squares The sum of squares is a measure of variance or deviation from the mean. It is calculated by summing of the squares of each score’s difference from the mean. Total sum of squares is another sum of squares; it considers not only the sum of squares from the factors, but also from randomness or error because the squaring of each score rids the equation of negative numbers. As you can see in the above example showing how to obtain the variance, step 5 requires you to find the sum of squares (SS). Useful definitions Measures of Variability: Measures that allow you to: determine the degree of variation within a population or sample; determine how representative a particular score is of a data set; and determine the scope and validity of any generalizations you wish to make based on your observations in research are measures of variability. Range: The range is the difference between the highest and lowest scores in a distribution Variability: degree to which the scores vary from their mean. Standard Deviation: Square root of the variance. Sum of Squares: The sum of squares is a measure of variance or deviation from the mean. It is calculated by summing of the squares of each score’s difference from the mean. It is the sum of squared deviations. Other Exercises and Resources For more information about or practice with measures of variability, see: H. Williams, 2015 3 Learning Centre MA 6.21 http://study.com/academy/lesson/skewed-distribution-examples-definitionquiz.html http://www.learningcommons.uoguelph.ca/guides/university_learning/handouts/hi ghlighting.pdf http://www.stats.ox.ac.uk/~marchini/teaching/L1/L1.slides.pdf http://davidmlane.com/hyperstat/intro.html http://study.com/academy/lesson/outlier-in-statistics-definition-lesson-quiz.html http://www.statisticshowto.com/skewed-distribution/ http://support.minitab.com/en-us/minitab/17/topic-library/modelingstatistics/anova/anova-statistics/understanding-sums-of-squares/ MEASURES OF VARIABILITY – ANSWER KEY 1. 8 11 12 14 17 17 18 19 22 29 35 38 M = (240/N) 240/12 = 20 Variance (8 – 20) = -12 (11 – 20) = -9 (12 – 20) = -8 (14 – 20) = -6 (17 – 20) = -3 (17 – 20) = -3 (18 – 20) = -2 (19 – 20) = -1 (22 – 20) = 2 (29 – 20) = 9 (35 – 20) = 15 (38 – 20) = 18 (-44) + 44 = 0 Variance = 81.83 SS (-12)2 = 144 (-9)2 = 81 (-8)2 = 64 (-6)2 = 36 (-3)2 = 9 (-3)2 = 9 (-2)2 = 4 (-1)2 = 1 22 = 4 92 = 81 152 = 225 182 = 324 982/12 =81.83 H. Williams, 2015 4